2 Specification

3 Description

To compute the inverse X of a real symmetric positive definite matrix A, F01ABF first computes a Cholesky factorization of A as A=LLT, where L is lower triangular. An approximation to X is found by computing L-1 and then the product L-TL-1. The residual matrix R=I-AX is calculated using additional precision, and a correction D to X is found by solving LLTD=R. X is replaced by X+D, and this iterative refinement of the inverse is repeated until full machine accuracy has been obtained.

4 References

5 Parameters

On entry: the upper triangle of the n by n positive definite symmetric matrix A. The elements of the array below the diagonal need not be set.

On exit: the lower triangle of the inverse matrix X is stored in the elements of the array below the diagonal, in rows 2 to n+1; xij is stored in Ai+1j for i≥j. The upper triangle of the original matrix is unchanged.

2: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F01ABF is called.

On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.

On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL=1

The matrix A is not positive definite, possibly due to rounding errors.

IFAIL=2

The refinement process fails to converge, i.e., the matrix A is ill-conditioned.